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Theorem acneq 8463
Description: Equality theorem for the choice set function. (Contributed by Mario Carneiro, 31-Aug-2015.)
Assertion
Ref Expression
acneq  |-  ( A  =  C  -> AC  A  = AC  C )

Proof of Theorem acneq
Dummy variables  f 
g  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2492 . . . 4  |-  ( A  =  C  ->  ( A  e.  _V  <->  C  e.  _V ) )
2 oveq2 6304 . . . . 5  |-  ( A  =  C  ->  (
( ~P x  \  { (/) } )  ^m  A )  =  ( ( ~P x  \  { (/) } )  ^m  C ) )
3 raleq 3023 . . . . . 6  |-  ( A  =  C  ->  ( A. y  e.  A  ( g `  y
)  e.  ( f `
 y )  <->  A. y  e.  C  ( g `  y )  e.  ( f `  y ) ) )
43exbidv 1758 . . . . 5  |-  ( A  =  C  ->  ( E. g A. y  e.  A  ( g `  y )  e.  ( f `  y )  <->  E. g A. y  e.  C  ( g `  y )  e.  ( f `  y ) ) )
52, 4raleqbidv 3037 . . . 4  |-  ( A  =  C  ->  ( A. f  e.  (
( ~P x  \  { (/) } )  ^m  A ) E. g A. y  e.  A  ( g `  y
)  e.  ( f `
 y )  <->  A. f  e.  ( ( ~P x  \  { (/) } )  ^m  C ) E. g A. y  e.  C  ( g `  y
)  e.  ( f `
 y ) ) )
61, 5anbi12d 715 . . 3  |-  ( A  =  C  ->  (
( A  e.  _V  /\ 
A. f  e.  ( ( ~P x  \  { (/) } )  ^m  A ) E. g A. y  e.  A  ( g `  y
)  e.  ( f `
 y ) )  <-> 
( C  e.  _V  /\ 
A. f  e.  ( ( ~P x  \  { (/) } )  ^m  C ) E. g A. y  e.  C  ( g `  y
)  e.  ( f `
 y ) ) ) )
76abbidv 2556 . 2  |-  ( A  =  C  ->  { x  |  ( A  e. 
_V  /\  A. f  e.  ( ( ~P x  \  { (/) } )  ^m  A ) E. g A. y  e.  A  ( g `  y
)  e.  ( f `
 y ) ) }  =  { x  |  ( C  e. 
_V  /\  A. f  e.  ( ( ~P x  \  { (/) } )  ^m  C ) E. g A. y  e.  C  ( g `  y
)  e.  ( f `
 y ) ) } )
8 df-acn 8366 . 2  |- AC  A  =  { x  |  ( A  e.  _V  /\  A. f  e.  ( ( ~P x  \  { (/)
} )  ^m  A
) E. g A. y  e.  A  (
g `  y )  e.  ( f `  y
) ) }
9 df-acn 8366 . 2  |- AC  C  =  { x  |  ( C  e.  _V  /\  A. f  e.  ( ( ~P x  \  { (/)
} )  ^m  C
) E. g A. y  e.  C  (
g `  y )  e.  ( f `  y
) ) }
107, 8, 93eqtr4g 2486 1  |-  ( A  =  C  -> AC  A  = AC  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1867   {cab 2405   A.wral 2773   _Vcvv 3078    \ cdif 3430   (/)c0 3758   ~Pcpw 3976   {csn 3993   ` cfv 5592  (class class class)co 6296    ^m cmap 7471  AC wacn 8362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-iota 5556  df-fv 5600  df-ov 6299  df-acn 8366
This theorem is referenced by:  acndom  8471  dfacacn  8560  dfac13  8561
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